The solution you suggest is to maintain ticket expenses in travel expense. It will be updated in FI when trip is settled and posted to FI.
In this condition the travel expense manager will settle the trip for paying the ticket cost to travel agencies before trip is completed by employee.
I don't think that's a good idea. If you do that, the status will be set to 3-0 prematurely, this will have consequences for your overall process cycle.
Once trip is completed by employee, will he able to edit the same trip for maintaining other expenses for settlement?
That depends entirely on the permissions you've assigned as per authorization object P_TRAVL. In general, this is possible, you can allow anybody to change any trip in any status (which would be nonsensial, but it's technically possible).
If I was you, I'd try to "calm" the accounting department and have them wait for the "clean" settlement of the trip, i.e. when it's really over and the expenses have been claimed, the travel manager has added the receipts manually or from CCC etc. Do you think that would be feasible in your business process?
The problem I see otherwise, is, that you will have to meddle with the standard process flow of FI-TV which will most probably mean more effort and cost than just coping with the slow speed of information transition to accounting.
Cheers, Lukas